Woo-Jin and Kiran were asked to find an explicit formula for the sequence $64\,,\,16\,,\,4\,,\,1,...$, where the first term should be $f(1)$. Woo-Jin said the formula is $f(n)=64\cdot\left(\dfrac{1}{4}\right)^{{n}}$, and Kiran said the formula is $f(n)=16\cdot\left(\dfrac{1}{4}\right)^{{n-1}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Woo-Jin (Choice B) B Only Kiran (Choice C) C Both Woo-Jin and Kiran (Choice D) D Neither Woo-Jin nor Kiran
Explanation: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{1}{4}=\dfrac{4}{16}=\dfrac{16}{64}={\dfrac{1}{4}}$ We see that the constant ratio between successive terms is ${\dfrac{1}{4}}$. In other words, we can find any term by starting with the first term and multiplying by ${\dfrac{1}{4}}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $f(n)$ ${64}\cdot\!\left({\dfrac{1}{4}}\right)^{0}$ ${64}\cdot\!\left({\dfrac{1}{4}}\right)^{1}$ ${64}\cdot\!\left({\dfrac{1}{4}}\right)^{2}$ ${64}\cdot\!\left({\dfrac{1}{4}}\right)^{3}$ We can see that every term is the product of the first term, ${64}$, and a power of the constant ratio, ${\dfrac{1}{4}}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${64}$ is the first term and ${\dfrac{1}{4}}$ is the constant ratio): $f(n)={64}\cdot\left({\dfrac{1}{4}}\right)^{{\,n-1}}$ We can now see that $f(n)=64\cdot\left(\dfrac{1}{4}\right)^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $f(1)=64\cdot\left(\dfrac{1}{4}\right)^{{\,1}} = 16$. However, according to our table of values, $f(1)=64$. So Woo-Jin is definitely wrong. What about Kiran? We can see that $f(n)=16\cdot\left(\dfrac{1}{4}\right)^{{\,n-1}}$ is also not a correct formula, because the first term according to this formula is $16$, while the actual first term is $64$. Hence, Kiran is also wrong. Neither Woo-Jin nor Kiran got a correct explicit formula.